We report the discovery of a warm Neptune and a hot sub-Neptune transiting TOI-421 (BD-14 1137, TIC 94986319), a bright (V = 9.9) G9 dwarf star in a visual binary system observed by the Transiting Exoplanet Survey Satellite (TESS) space mission in Sectors 5 and 6. We performed ground-based follow-up observations—comprised of Las Cumbres Observatory Global Telescope transit photometry, NIRC2 adaptive optics imaging, and FIbre-fed Echellé Spectrograph, CORALIE, High Accuracy Radial velocity Planet Searcher, High Resolution Échelle Spectrometer, and Planet Finder Spectrograph high-precision Doppler measurements—and confirmed the planetary nature of the 16 day transiting candidate announced by the TESS team. We discovered an additional radial velocity signal with a period of five days induced by the presence of a second planet in the system, which we also found to transit its host star. We found that the inner mini-Neptune, TOI-421 b, has an orbital period of Pb = 5.19672 ± 0.00049 days, a mass of Mb = 7.17 ± 0.66 M⊕, and a radius of Rb = ${2.68}_{-0.18}^{+0.19}$ R⊕, whereas the outer warm Neptune, TOI-421 c, has a period of Pc = 16.06819 ± 0.00035 days, a mass of Mc = ${16.42}_{-1.04}^{+1.06}$ M⊕, a radius of Rc = ${5.09}_{-0.15}^{+0.16}$ R⊕, and a density of ρc = ${0.685}_{-0.072}^{+0.080}$ g cm-3. With its characteristics, the outer planet (ρc = ${0.685}_{-0.072}^{+0.080}$ g cm-3) is placed in the intriguing class of the super-puffy mini-Neptunes. TOI-421 b and TOI-421 c are found to be well-suited for atmospheric characterization. Our atmospheric simulations predict significant Lyα transit absorption, due to strong hydrogen escape in both planets, as well as the presence of detectable CH4 in the atmosphere of TOI-421 c if equilibrium chemistry is assumed. * Based on observations made with ESO Telescopes at the La Silla Observatory under programs ID 1102.C-0923, 0103.C-0874, 0103.C-0759, 0103.C-0442, and 60.A-970. Based on observations obtained with the Nordic Optical Telescope (NOT), operated on the island of La Palma jointly by Denmark, Finland, Iceland, Norway, and Sweden, in the Spanish Observatorio del Roque de los Muchachos (ORM) of the Instituto de Astrofísica de Canarias (IAC). This paper includes data gathered with the 6.5 m Magellan Telescopes located at Las Campanas Observatory, Chile. This work makes use of observations from the LCOGT network.